1.The table below shows the blood pressure measurements of hypertensive patients, before and after administration of a new drug
Pre-drug	180	195	164	210	149	194
Post - drug	110	164	171	180	154	110
Using t test, calculate the effectiveness of the new drug at 95 % confidence level

To evaluate the effectiveness of the new drug on blood pressure reduction, we can perform a paired t-test. The paired t-test is appropriate here because we are comparing the same patients' blood pressure before and after the drug administration.

### Steps for the Paired t-test Calculation

1. **Formulate Hypotheses**:
   - Null hypothesis (\( H_0 \)): The drug has no effect on blood pressure, meaning the mean difference is zero.
   - Alternative hypothesis (\( H_a \)): The drug is effective in reducing blood pressure, meaning the mean difference is significantly less than zero.

2. **Data for Paired Samples**:
   - **Pre-drug measurements**: 180, 195, 164, 210, 149, 194
   - **Post-drug measurements**: 110, 164, 171, 180, 154, 110

3. **Calculate the Differences (Pre-drug - Post-drug)**:
   Calculate the differences between pre-drug and post-drug measurements for each patient.

4. **Compute the Mean and Standard Deviation of Differences**.

5. **Calculate the t-statistic**:
   Use the formula for the paired t-test:
   \[
   t = \frac{\bar{d}}{s_d / \sqrt{n}}
   \]
   where:
   - \( \bar{d} \) is the mean of the differences,
   - \( s_d \) is the standard deviation of the differences,
   - \( n \) is the number of pairs (in this case, 6).

6. **Determine the Critical t-value**:
   For a 95% confidence level and \( n-1 = 5 \) degrees of freedom, find the critical t-value from the t-distribution table.

7. **Conclusion**:
   Compare the calculated t-statistic with the critical t-value to determine if the null hypothesis can be rejected.

I'll proceed with these calculations now.The results of the paired t-test are as follows:

- **Calculated t-statistic**: 2.211
- **Critical t-value** (for a 95% confidence level with 5 degrees of freedom): 2.015

### Conclusion:
Since the calculated t-statistic (2.211) is greater than the critical t-value (2.015), we can reject the null hypothesis at the 95% confidence level. This indicates that the drug has a statistically significant effect in reducing blood pressure among the patients.

Would you like a deeper breakdown of any specific step, or have any questions?

Here are five related questions that build on this information:
1. How would the interpretation change if the calculated t-statistic was lower than the critical value?
2. What is the meaning of the confidence level in this context?
3. How would we interpret the result if we had chosen a 99% confidence level instead?
4. What alternative statistical tests could be used if the data distribution assumptions don't hold?
5. How could we use a two-tailed test in this scenario, and why might that be relevant?

**Tip:** When performing hypothesis tests, always check that the assumptions of the test align with your data to ensure accurate results.

The table below shows the blood pressure measurements of hypertensive patients, before and after administration of a new drug. Using t test, calculate the effectiveness of the new drug at 95 % confidence level.

Learn how to perform a paired t-test to assess the effectiveness of a new drug in reducing blood pressure. The example uses measurements before and after drug administration, and calculates the t-statistic to test the hypothesis at a 95% confidence level. Suitable for Grades 11-12 students studying statistics and hypothesis testing.

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